Quantization of rotating linear dilaton black holes

............. manuscript No. (will be inserted by the editor)

Quantization of rotating linear dilaton black holes

arXiv:1406.5130v2 [gr-qc] 24 Jun 2014

I. Sakalli1 Department of Physics, Eastern Mediterranean University, Gazimagosa, North Cyprus, Mersin 10, Turkey. e-mail: [email protected]

The date of receipt and acceptance will be inserted by the editor

Abstract

In this paper, we firstly prove that the adiabatic invariant quan-

tity, which is commonly used in the literature for quantizing the rotating black holes (BHs) is fallacious. We then show how its corrected form should be. The main purpose of this paper is to study the quantization of 4-dimensional rotating linear dilaton black hole (RLDBH) spacetime that is describing with an action, which emerges in the Einstein-Maxwell-DilatonAxion (EMDA) theory. The RLDBH spacetime has a non-asymptotically flat (NAF) geometry. They reduces to the linear dilaton black hole (LDBH) metric when vanishing its rotation parameter ”a”. While studying its scalar perturbations, it is shown that the Schr¨odinger-like wave equation around the event horizon reduces to a confluent hypergeometric differential equation. Then the associated complex frequencies of the quasinormal modes (QNMs) are computed. By using those QNMs in the true definition of the rotational adiabatic invariant quantity, we obtain the quantum spectra of Send offprint requests to:

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I. Sakalli

entropy/area for the RLDBH. It is found out that both spectra are discrete and equidistant. Besides, we reveal that the quantum spectra do not depend on a in spite of the QNMs are modulated by it.

Key words

Rotating Linear Dilaton Black Holes, Quasinormal Modes,

Entropy/Area Spectra, Confluent Hypergeometric Function, Axion, Zerilli Equation.

1 Introduction Quantization of BHs has always attracted major interest for the scientists who deal with quantum gravity theory. The onset of this topic dates back to 1970s, in which Bekenstein who firstly proved that BH entropy is proportional to the area of the BH horizon [1,2] and found an equidistant area spectrum [3,4,5] via

An = 8πξ~ = ǫnlp2 ,

(n = 0, 1, 2.......),

(1)

where An denotes the area spectrum of the BH horizon, n is the associated quantum number, ξ represents a number that is order of unity, ǫ is a dimensionless constant and lp stands for the Planck length. In his celebrated studies, Bekenstein considered the BH horizon area as an adiabatic invariant quantity, and used the Ehrenfest’s principle in order to obtain the above discrete and equally spaced area spectrum. It is clear from Eq. (1) that when a test particle is swallowed by a BH, the minimum increase in the horizon area would be ∆Amin = ǫlp2 . Meanwhile, throughout the paper, we follow-up the fundamental units with c = G = 1 and lp2 = ~. According to Bekenstein [3], a BH horizon is made by patches of equal area ǫ~ with ǫ = 8π. Up to the present, many attempts have been made for deriving

Quantization of rotating linear dilaton black holes

3

such evenly spaced area spectrum. On the other hand, depending on the which method employed for getting the spectrum, the value of ǫ has been somewhat controversial (for the topical review, a reader may refer to [6] and references therein). QNMs, known as the characteristic ringing of BHs, consist of damped oscillations that can be characterized by a discrete set of complex frequencies. They are determined by the solution of the wave equation when a BH is perturbed by an exterior field. Furthermore, a perturbed BH tends to equilibrate itself by emitting energy in the form of gravitational waves. Because of that QNMs are also important for observational aspect of gravitational waves phenomena, e.g. LIGO [7]. Taking inspiration from the latter remarks, firstly Hod [8,9] hypothesized that ǫ can be computed by utilizing the QNM of a vibrating BH. To this end, Hod used the Bohr’s correspondence principle [10] and conjectured that the real part of the asymptotic QNM frequency (Re ω) of a highly damped BH is relevant to the quantum transition energy between sequential quantum levels of the BH. Thus, this vibrational frequency engenders a change in the BH mass as ∆M = ~ (Re ω). For the Schwarzschild BH, Hod’s calculations resulted in ǫ = 4 ln 3. Afterwards, Kunstatter [11] used the natural adiabatic invariant quantity Iadb , which is defined for a system with energy E as

Iadb =

Z

dE , ∆ω

(2)

where ∆ω = ωn−1 − ωn represents the transition frequency. At large quantum numbers (n → ∞), Iadb behaves as a quantized quantity (Iadb ≃ n~) due to the Bohr-Sommerfeld quantization condition. After calculating ∆ω, by using the Re ω, and replacing E with the mass M of the static BH, Kunstatter managed to reproduce the Hod’ result ǫ = 4 ln 3 for the

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I. Sakalli

Schwarzschild BH. In 2008, Maggiore [12] suggested another method in which the proper physical frequency of the harmonic oscillator with a damp√ ing term should be in the form of ω = Re ω2 + Im ω 2 , where Im ω denotes the imaginary part of the QNM frequency. Actually, this form of the proper physical frequency for the QNM frequency was firstly employed in [13]. So, when Re ω ≫ Im ω one can obtain the Hod’s result. However, in the case of highly excited QNMs, it is a known fact that Im ω ≫ Re ω and consequently ∆ω ≈ Im ωn−1 − Im ω n that is in general directly proportional to the Hawking temperature T of the BH. With this new identification, for the Schwarzschild BH, it was found that ǫ = 8π, which is nothing but the original result of Bekenstein. By now, Maggiore’s method (MM) has been widely studied by the other researchers for the various BH backgrounds (see for instance [14,15,16,17,18,19,20,21,22]). Thereafter, Vagenas [23] and Medved [24] amalgamated the results of Kunstatter and Maggiore, and claimed that rotational version of the adiabatic invariant should be in the following form.

rot Iadb =

Z

dM − ΩdJ = n~, ∆ω

(3)

where Ω and J represent angular velocity and angular momentum of the rotating BH, respectively. In [23,24], asymptotic form of the quasinormal modes of the Kerr BH [25] were used in the above expression and an equidistant area spectrum ”under the condition of M 2 ≫ J” was obtained. Namely, if the associated condition fails, there is a non-equidistant spectrum, and hence the conundrum with Bekenstein’s result (1) arises. As for me, the definition (3) given by Vagenas and Medved is fallacious. I shall now prove this statement rigorously. Let us first note that the above

Quantization of rotating linear dilaton black holes

5

rot definition of Iadb used by Vagenas and Medved is originated from the first

law of thermodynamics:

Z T dS 1 , 4 T Z Z dM − ΩdJ T dS = = n~, ≈ ∆ω ∆ω

rot Iadb =

Z

dA =

1 4

Z

dS =

(4)

in which S shows the entropy of the BH. But, while doing this, they made a basic mathematical mistake. Because, the derivation shown above should be corrected as follows:

rot dIadb = dA ≈

T dS dM ΩdJ = − = 0, ∆ω ∆ω ∆ω

(5)

which is nothing but the homogeneous type of first-order differential equation. For the exactness,

∂ ∂J



1 ∆ω



∂ = ∂M



−Ω ∆ω



,

(6)

must hold. Then, the solution of Eq. (4), which yields the Iadb for the rotating BHs is (see for instance [26]).

rot Iadb =

Z

dM − ∆ω

Z 

Ω ∂ + ∆ω ∂J

Z

dM ∆ω



dJ = n~,

(7)

Now, we can test it on the Kerr BH. If one substitutes the values of (Ω, J, ∆ω) of the Kerr BH, as being used in [23,24], into Eq. (7), we obtain

rot Iadb = M2 +

p An = n~ M4 − J2 = 8π

→ ∆Amin = 8π~,

(8)

which is fully agreement with the Bekenstein’s result. However, if the same calculation was done with the definition (3), we would find out

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I. Sakalli

rot Iadb = M2 + 2

p An , M 4 − J 2 6= 8π

(9)

Obviously, it does not yield an equidistant area spectrum. The latter remark also explains the ambiguity appeared in the results of [23,24]. In the present study, we aim to analyze the entropy/area spectra of the RLDBHs. The eponyms of the RLDBHs are Cl´ement and Gal’tsov [27]. They are NAF solutions to the EMDA theory in 4-dimensions. Although there are many studies in the literature on their non-rotating form (known as LDBH) including subjects of Hawking radiation, entropic force, higher dimensions, quantization and the lensing effect on the Hawking temperature [28,29,30,31,32,33,34,35,36,37], the studies on the RLDBH remained very limited. According to our knowledge, the whole studies on the RLDBH are these references [38,39,40], which consist branes, holography, greybody factor, Hawking radiation and QNMs. On the other hand, up to date the problem of the quantization of the RLDBH has not been studied, yet. So, we want to fill this gap in the literature. Meanwhile, the method which we are going to follow here for calculating the QNMs of the RLDBH differs from the one used in [40]. Actually, Eq. (45) of [40] gives only one of the two sets of the QNMs of the RLDBHs and it is independent of the rotation parameter a. However, the general form of the QNMs of the RLDBH must include the rotation term, a. In fact, if one uses bs = −n (see Eq. (17) of [40]) instead of as = −n for evaluating the QNMs, it would be possible to derive the yet another set of the QNMs with a. Thus, while a → 0, one can read the QNMs of the LDBH from that result. Somehow this point has been disregarded in [40]. Our method for having the QNMs is based on the former studies [41, 42], which use an approximation method that considers small perturbations

Quantization of rotating linear dilaton black holes

7

around the event horizon of a BH. This method predicates on the termination of the outgoing waves at the horizon. To do this, it uses the poles of gamma function. The most recent works in the same line of thought can be seen in [34,35,43,44,45]. We analytically show how one gets the QNMs of the RLDBHs in terms of the angular velocity, as in the case of the Kerr BH [25]. Then we proceed to derive equally spaced entropy/area spectra for the RLDBHs. The remainder of the paper is arranged as follows. Sec. 2 is devoted to the introduction of the RLDBH metric. The Klein-Gordon (KG) equation for a massless scalar field is analyzed in this geometry. We also show how the separation of variables renders possible to reduce the physical problem to a Schr¨odinger-type wave equation, which is the so-called Zerilli equation [46]. In Sec. 3, it is shown that in the vicinity of the event horizon the Zerilli equation is reduced to a confluent hypergeometric differential equation. Then, we show how one reads the QNMs of the RLDBHs, and how they are being used to obtain the entropy/area spectra of the RLDBHs. Finally, we finish with the conclusions in Sec. 4.

2 RLDBHs and KG fields

In this section, we make the general overview of the RLDBH solution and serve its basic thermodynamical features. Then, we study the KG equation in the RLDBH background. After separating the radial equation, we also show how it reduces to the Zerilli equation . The action of the EMDA theory, which comprises the dilaton field φ and the axion (pseudoscalar) κ coupled to an Abelian vector field A is given by

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I. Sakalli

1 S= 16π

Z

  p 1 4φ µν µ −2φ µν µ e , Fµν F − κFµν F d x |g| −R + 2∂µ φ∂ φ + e ∂µ κ∂ κ − e 2 (10) 4

where Fµυ is the Maxwell two-form associated with a U (1) subgroup of E8 × E8 or Spin(32)/Z2 , and Feµν represents the dual of Fµυ . In the absence of the NUT charge, the RLDBH solution was found by [27] as follows "  2 # dr2 a 2 2 ds = −f (r)dt + + R(r) dθ + sin θ dϕ − dt , f (r) R(r) 2

2

(11)

where the metric functions are given by R(r) = rr0 ,

f (r) =

(12)

Λ , R(r)

(13)

where Λ = (r − r2 )(r − r1 ) in which r1 and r2 represent the inner and outer (event) horizons that are derived from the condition f (r) = 0. These radii are given by

ri = M + (−1)i

p M 2 − a2 ,

(i = 1, 2),

(14)

where M is an integration constant, which is related to the quasilocal mass (MQL ) of the RLDBH and as mentioned before a denotes the rotation parameter determining the angular momentum of the RLDBH. The background electric charge Q is related with the constant parameter r0 via √ 2Q = r0 . Furthermore, the background fields are given by

e−2φ =

R(r) , r2 + a2 cos2 θ

(15)

Quantization of rotating linear dilaton black holes

κ=−

r2

r0 a cos θ , + a2 cos2 θ

9

(16)

and the Maxwell field (F = dA) is derivable from the following electromagnetic four-vector potential 1 A = √ (e2φ dt + a sin2 θdϕ). 2

(17)

From Eq. (14), we infer that

M ≥ a,

(18)

for having a BH. Meanwhile, since the RLDBH has a NAF geometry, one could follow the MQL definition of Brown and York [47]. Thus, we obtain

MQL =

M . 2

(19)

Besides, the angular momentum J is computed as

J=

ar0 . 2

(20)

According to the definition given by Wald [48], the surface gravity of the RLDBH can simply be computed through the following expression:

κ=

(r2 − r1 ) f ′ (r) = , 2 r=r2 2r2 r0

(21)

where a prime ”′” denotes differentiation with respect to r. From hence, we obtain the Hawking temperature [48,49] of the RLDBH as

~κ , 2π ~ (r2 − r1 ) . = 4πr2 r0

TH =

(22)

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I. Sakalli

As it can be seen from above, in the extremal case (M = a) the temperature vanishes since r2 = r1 . The Bekenstein-Hawking entropy of a BH in Einstein theory is defined as

SBH =

πr2 r0 Ah = , 4~ ~

(23)

where Ah is area of the surface of the event horizon. In order to satisfy the first law of BH mechanics, we also need the angular velocity which takes the following form for the RLDBH:

ΩH =

a . r2 r0

(24)

With these definitions, the validity of the first law of thermodynamics for the RLDBH can be proven via

dMQL = TH dSBH + ΩH dJ.

(25)

It is worth mentioning that the above identity does not include Q. Because, here the electric charge Q is considered as a background charge which has a fixed value. This feature exhibits one of the characteristics of the linear dilaton backgrounds [27]. In order to obtain the entropy spectrum of the RLDBH via the MM, here we firstly consider the massless KG equation on that geometry. In general, the massless KG equation in a curved spacetime is given by √ ∂j ( −g∂ j Φ) = 0,

j = 0...3.

(26)

We invoke the following ansatz for the scalar field Φ in the above equation ℜ(r) Φ = √ e−iωt Ylm (θ, ϕ), r

Re(ω) > 0,

(27)

Quantization of rotating linear dilaton black holes

11

in which Ylm (θ, ϕ) is the well-known spheroidal harmonics which admits the eigenvalue −l(l + 1) [50]. Here, m and l represent the magnetic quantum number and orbital angular quantum number, respectively. Thus one can obtain the following radial equation # " 2 ′ Λ 3Λ (ωrr − ma) Λ 0 ℜ(r) = 0, − l(l + 1) + 2 − Λℜ′′ (r) + (Λ′ − )ℜ′ (r) + r Λ 4r 2r (28) From this, we can obtain the Zerilli equation [46] as follows 

−

d2 +V dr∗2



ℜ(r∗ ) = ω2 ℜ(r∗ ),

(29)

in which the effective potential is computed as

V = f (r) where

(

)  2  e 1 r + 2M r − 3a2 mΩ e − 2ω) , (30) + l(l + 1) − (mΩ R(r) 4r2 f (r)

e= a . Ω rr0

(31)

The tortoise coordinate r∗ is defined as

∗

r = which leads to

Z

dr , f (r)

# " r ( r2 − 1)r2 r0 . ln r = r2 − r1 (r − r1 )r1 ∗

(32)

(33)

Finally, one can easily check that the asymptotic limits of r∗ take the following form

lim r∗ = −∞ and

r→r2

lim r∗ = ∞.

r→∞

(34)

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I. Sakalli

3 QNMs and entropy/area spectra of RLDBHs

The QNMs are defined as the solutions to (29) that satisfy the casual condition of purely ingoing wave at the event horizon and purely outgoing wave at the spatial infinity. However, this description is conventionally made for the asymptotically flat BHs, whose their potentials terminate at the spatial infinity (for instance, one may recall the Zerilli potentials of the Schwarzschild and Kerr BHs [46]). On the other hand, it can be checked that the potential (30) of our NAF BH (with non-oversized charge, Q 6= ∞) does not vanish at the spatial infinity (limr∗ →∞ V =

1 1 [ r02 4

+ l(l + 1)]), even for the s-waves

(l = 0). Furthermore, it may diverge for ultrahigh orbital quantum numbers (l → ∞). In this section, we follow up the recent studies [34,35,43,44,45] in which the particular approximation method [41,42] is employed in order to compute the QNMs. This method is based on the poles of the scattering amplitude, which occurs when the argument of the Gamma function take negative integer. According to this method, the datum point for computing the QNM frequencies is to impose the condition of ”the QNMs should be purely ingoing plane wave at the event horizon”:

∗

ℜ(r)|QN M ∼ eiωr ,

(r∗ → −∞).

(35)

In order to employ the associated method, we first use the Taylor series expansion of the metric function f (r) with respect to r around the expansion point r2 :

Quantization of rotating linear dilaton black holes

13

f (r) ≃ f ′ (r2 )(r − r2 ) + a(r − r2 )2 , ≃ 2κ(r − r2 ),

(36)

≃ 2κx, where x = r − r2 . So the tortoise coordinate transforms into

r∗ ≃

1 ln x. 2κ

(37)

Thus, one can write the near horizon form of the potential (30) as

Vhorizon ≃ 2κx



   x κx mΩH 1 l(l + 1) − κ (1 − ) − 2 − (mΩH − 2ω) . r2 r0 r2 2r2 2κx (38)

While doing this, Taylor expansion around x = 0 is performed. Substituting Eqs. (37) and (38) into Eq. (29), we reduce the Zerilli equation to the following differential equation

d2 ℜ(x) x2 dx2


ω 2 − Vhorizon dℜ(x) +x ℜ(x) = 0. + dx 4κ2

(39)

Its solution can be found in terms of the Whittaker functions [51] as follows.

i 1 h a, eb; ie cx) + C2 W hittakerW (e a, eb; ie cx) , ℜ(x) = √ C1 W hittaker(e x where C1 and C2 are the integration constants, and

(40)

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I. Sakalli

µ , e a=i √ λ κr0 b eb = i ω , 2 λ , e c= √ 2r2 κr0

in which

ω b=

(41)

mΩH − ω , κ

µ = l(l + 1) + κr0 , p λ = 2 2l(l + 1) + 3κr0 .

(42)

However, our calculations showed that W hittakerW function seen in Eq. (40) admits exponentially growth modes near the event horizon, instead of the damping modes . For this reason, we set C2 = 0. From the mathematical point of view, the necessity of setting C2 = 0 is related to the branch cut discontinuity of the generalized Laguerre polynomial (for the details, see [44]). Then the near horizon solution of the radial equation can be expressed in terms of the confluent hypergeometric functions [51] as

1 e ℜ(x) ∼ xb U ( − e a + eb, 1 + 2eb; ie cx), 2

(43)

In the limit of x ≪ 1, the above solution takes the following form e

ℜ(x) ∼ D1 x−b

Γ (2eb)

a + eb) Γ ( 12 − e

e

+ D2 xb

Γ (−2eb)

a − eb) Γ ( 21 − e

,

(44)

where constants D1 and D2 stand for the amplitudes of the near-horizon outgoing and ingoing waves, respectively. We infer from the boundary condition (35) that the first term of Eq. (44) must terminate. The pole of the

Quantization of rotating linear dilaton black holes

15

Gamma function Γ ( 21 − e a + eb) renders possible of that termination. Thus,

we obtain

2µ + i(2s + 1), ω bs = √ λ κr0

(s = 0, 1, 2, ...)

(45)

So, with the aid of Eq. (41) one can derive the QNM frequencies as 2µ ωs = mΩH − λ

r

κ − i(2s + 1)κ r0

(46)

We can easily check that the above result is in full agreement with the results obtained for the 4-dimensional non-rotating (a = 0 or r1 = 0) LDBHs [34,35]. Thus, we conclude that the imaginary part of the frequency of the QNM of the RLDBH is given by

Im ω s = −(2s + 1)κ,

1 r2 − r1 , = −(s + ) 2 r2 r0

(47)

Therefore, we can find the transition frequency between two highly damped neighboring states as follows

∆ω ≈ Im ω s−1 − Im ωs , =

r2 − r1 , r2 r0

(s → ∞) (48)

= 4πTH /~. After substituting the above result into Eq. (7), one can straightforwardly evaluate the integrals and obtain

rot Iadb =

Ah r2 r0 = = n~. 4 16π

From above, one reads the area spectrum as

(49)

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I. Sakalli

An = 16πn~,

(50)

which measures the minimum area spacing as

∆Amin = 16π~. Since S =

A 4~ ,

(51)

the entropy spectrum can also be obtained as

Sn = 4πn.

(52)

The above results represent that the RLDBHs have equidistant quantum spectrum. Furthermore, it is easily seen that the spectroscopy of the RLDBH is completely independent of the rotation parameter a.

4 Conclusion In this paper, we have first revisited the definition of the rotational adiabatic invariant quantity. Secondly, we have corrected its erroneous formulation. Then, as an example of its usage, we have considered the RLDBH geometry. According to the MM, which considers the perturbed BH as a highly damped harmonic oscillator, the transition frequency in the expression (7) rot of Iadb is governed by Im ω rather than Re ω of the QNMs. For comput-

ing Im ω, we have used the near-horizon form of the Zerilli equation (29), and achieved to express it as a confluent hypergeometric differential equation (39). After a straightforward calculation, the parameters of the confluent hypergeometric function (43) are rigorously found. Then, we have computed the complex QNM frequencies of the RLDBH that possess the LDBH limit (a → 0) [34,35]. After getting the transition frequency for the

Quantization of rotating linear dilaton black holes

17

highly damped QNMs, we have obtained the quantized entropy/area spectra of the RLDBH. Both spectra are irrelevant to the rotation parameter a and equally spaced. Although the RLDH is a stationary BH, our result is consistent with the Kothawala et al.’s conjecture [52], which states that the spacing of the area spectrum depends on the associated gravity theory; in Einstein’s gravity it is equally spaced, otherwise (in higher-derivative gravity theory) it is not. On the other hand, the value of the numerical coefficient is found to be ǫ = 16π or ξ = 2. Although this result matches with the previous results [34,35,43,44,45], it shows incongruity with the Bekenstein’s well-known value, ξ = 1. However, as stated in [9], in the spectroscopy calculations having equi-spacing is the foremost issue. The equi-spacings between two neighboring levels may vary depending on the which method is used. Finally, we plan to extend the same analysis to the Dirac equation which can be expressed in the context of the Newman-Penrose formalism [46] in order to analyze the quantization of the stationary spacetimes [53,54] with the aid of the QNMs of fermions. This is going to be our next problem in the near future.

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